sketch the curve for each pair of parametric equations.\n37) $x = t, y=\frac{t^{2}}{5}$
sketch the curve for each pair of parametric equations.\n37) $x = t, y=\frac{t^{2}}{5}$
Answer
Explanation:
Step1: Substitute (t) with (x)
Since (x = t), we can substitute (t) in the (y) - equation. So (y=\frac{x^{2}}{5}).
Step2: Identify the type of curve
The equation (y = \frac{1}{5}x^{2}) is a parabola. The general form of a parabola is (y = ax^{2}+bx + c), and in our case (a=\frac{1}{5}), (b = 0), (c = 0). The vertex of the parabola (y=ax^{2}+bx + c) is at (x=-\frac{b}{2a}). Since (b = 0), the vertex is at (x = 0). When (x = 0), (y=0), so the vertex is ((0,0)). Also, since (a=\frac{1}{5}>0), the parabola opens upwards.
Step3: Find some points
Choose some values of (x) to find corresponding (y) - values. When (x=- 5), (y=\frac{(-5)^{2}}{5}=\frac{25}{5}=5). When (x = 5), (y=\frac{5^{2}}{5}=5). When (x=-2), (y=\frac{(-2)^{2}}{5}=\frac{4}{5}=0.8). When (x = 2), (y=\frac{2^{2}}{5}=0.8).
Answer:
Sketch a parabola with vertex at the origin ((0,0)) opening upwards, passing through points such as ((-5,5)), ((-2,0.8)), ((0,0)), ((2,0.8)), ((5,5)) etc.